Characteristic cycles of standard modules for the rational Cherednik algebra of type Z/lZ

Abstract

We study the representation theory of the rational Cherednik algebra $H_\kappa = H_\kappa({\mathbb Z}_l)$ for the cyclic group ${\mathbb Z}_l = {\mathbb Z} / l {\mathbb Z}$ and its connection with the geometry of the quiver variety $M_\theta(\delta)$ of type $A_{l-1}^{(1)}$. We consider a functor between the categories of $H_\kappa$-modules with different parameters, called the shift functor, and give the condition when it is an equivalence of categories. We also consider a functor from the category of $H_\kappa$-modules with good filtration to the category of coherent sheaves on $M_\theta(\delta)$. We prove that the image of the regular representation of $H_\kappa$ by this functor is the tautological bundle on $M_\theta(\delta)$. As a corollary, we determine the characteristic cycles of the standard modules. It gives an affirmative answer to a conjecture given in [Gordon, arXiv:math/0703150v1] in the case of ${\mathbb Z}_l$.